1. Technical Field
The present disclosure relates to the field of switched charge storage element networks and, more specifically, to switched charge storage element integrators.
2. Description of the Related Art
Switched charge storage element networks i.e., switched capacitor networks are widely used to perform several functions. One application of a switched capacitor network is sigma delta modulators. Sigma delta modulators encode high resolution signals into low resolution signals using pulse-density modulation, and they are used in various modern electronic devices, such as analog-to-digital and digital-to-analog converters, frequency synthesizers, switched-mode power supplies, and motor controls. There are predominantly two approaches for realizing sigma delta modulators, namely, discrete time architecture and continuous time architecture. Discrete time modulators have some advantages over their continuous time counterparts in terms of robustness with process variation, tolerance towards clock jitter, and feasibility to cascade multiple modulators to form multistage (MASH) architecture. However, discrete time modulators being sampled data systems require an anti-aliasing filter, which consumes substantial amount of silicon area. The continuous time modulators do not require an anti-aliasing filter and hence are a promising proposition for low area solution. However, continuous time modulators suffer from limitations of clock jitter sensitivity and rise/fall transients of feedback DAC. To address the issues arising as above, a hybrid of continuous time and discrete time architectures provides a discrete time switched capacitor DAC that replaces the continuous time feedback DAC in the modulator.
FIG. 1 illustrates a conventional second order sigma delta (ΣΔ) modulator 100. The ΣΔ modulator 100 includes two integrators 101, a quantizer and feedback DACs. The ΣΔ modulator also includes two subtractors to form the basic building block.
FIG. 2 illustrates a conventional schematic diagram of an integrator 200. The integrator 200 includes an operational amplifier XOPA, capacitors (Ci, C1), resistors (Ri, R1), and switches (S1, S2, S3, S4). Integrating capacitor Ci is coupled between the input terminal INM and the output node OUT of operational amplifier XOPA. The reference voltage node VCM coupled to the input terminal INP, acts as small signal analog ground. Resistor Ri is coupled between terminal INM and an analog input node VIN. The top plate of capacitor C1 is coupled to terminal INM through a switch S1 that switches “ON” during phase PH1 active. The top plate is also coupled to reference voltage node VCM through switch S2 which switches “ON” during phase PH2 active. The bottom plate of C1 is coupled to reference voltage node VCM through series resistor R1 and switch S3, which switches “ON” during phase PH1 active. The bottom plate is also connected to the output of a local DAC through switch S4, which switches “ON” during phase PH2 active. In particular, during phase PH2 active, top plate of capacitor C1 is coupled to reference voltage VCM, while its bottom plate samples the DAC output. During phase PH1 active, the top plate of C1 is coupled to the input terminal INM of the operational amplifier while its bottom plate is coupled to VCM through resistor R1. Hence during phase PH1 active, C1 transfers a charge approximating C1*VDACOUT to the integrating capacitor Ci, where VDACOUT is the output voltage of the feedback DAC.
The time period of phase PH1 and phase PH2 is denoted as T and rising edge of PH2 is assumed as the beginning of a sample phase in the rest of the background disclosure.
Assuming R1*Ci<<T at the end of sample phase ‘n’, the output of integrator is approximated as:
                                          V            OUT                    ⁡                      [            n            ]                          =                                            (                                                C                  1                                                  C                  i                                            )                        ×                                          ∑                                  i                  =                  1                                n                            ⁢                                                V                  DACOUT                                ⁡                                  [                  i                  ]                                                              +                                    1                              (                                                      R                    i                                    ⁢                                      C                    i                                                  )                                      ⁢                          ∫                                                                    V                    IN                                    ⁡                                      (                    t                    )                                                  ⁢                                  ⅆ                  t                                                                                        (        1        )            
Equation (1) denotes the basic operation of the integrator used inside a continuous time sigma delta modulator with discrete time feedback.
FIG. 3 illustrates an integrator circuit 300 equivalent to the conventional integrator 200 during the phase PH2 active. The capacitor C1, shown in FIG. 2, is not coupled to the terminal INM during phase PH2 active and hence has been removed from FIG. 3. The supply noise is introduced by means of a random noise source Vn(t) applied at positive input terminal INP of the operational amplifier XOPA.
By mathematical manipulation it is clear that the equivalent noise source referred at VIN during phase PH2 is approximated by the equation:
                                          V                          neqph              ⁢                                                          ⁢              2                                ⁡                      (            t            )                          =                                            V              n                        ⁡                          (              t              )                                +                                    (                                                R                  i                                ⁢                                  C                  i                                            )                        ⁢                                          ⅆ                                                                                              ⅆ                t                                      ⁢                                          V                n                            ⁡                              (                t                )                                                                        (        2        )            
FIG. 4 illustrates the integrator circuit 400 equivalent to the conventional integrator 200 during the phase PH1 active. Switch 51 is ON and couples the top terminal of capacitor C1 to terminal INM. The bottom plate of capacitor C1 is coupled to the reference voltage VCM.
By mathematical manipulation, the equivalent noise source at VIN during phase PH1 is:
                                          V                          neqph              ⁢                                                          ⁢              1                                ⁡                      (            t            )                          =                                                            V                n                            ⁡                              (                t                )                                      ×                          (                              1                +                                                      C                    i                                    /                                      C                    1                                                              )                                +                                    (                                                R                  i                                ⁢                                  C                  i                                            )                        ⁢                                          ⅆ                                                                                              ⅆ                t                                      ⁢                                          V                n                            ⁡                              (                t                )                                                                        (        3        )            
Using equations (2) and (3):
Total equivalent noise at VIN is:
                                                        V              neq                        ⁡                          (              t              )                                =                                                                      V                                      neqph                    ⁢                                                                                  ⁢                    2                                                  ⁡                                  (                  t                  )                                            ×                              U                ⁡                                  (                                      PH                    ⁢                                                                                  ⁢                    2                                    )                                                      +                                                            V                                      neqph                    ⁢                                                                                  ⁢                    1                                                  ⁡                                  (                  t                  )                                            ×                              U                ⁡                                  (                                      PH                    ⁢                                                                                  ⁢                    1                                    )                                                                    ⁢                                  ⁢                              U            ⁡                          (                              PH                ⁢                                                                  ⁢                2                            )                                =                                    0              ⁢                                                          ⁢              when              ⁢                                                          ⁢              PH              ⁢                                                          ⁢              2              ⁢                                                          ⁢              is              ⁢                                                          ⁢              LOW                        ⁢                                                  ⁢                                                  =                          1              ⁢                                                          ⁢              when              ⁢                                                          ⁢              PH              ⁢                                                          ⁢              2              ⁢                                                          ⁢              is              ⁢                                                          ⁢              HIGH                                      ⁢                                  ⁢                              U            ⁡                          (                              PH                ⁢                                                                  ⁢                1                            )                                =                                    0              ⁢                                                          ⁢              when              ⁢                                                          ⁢              PH              ⁢                                                          ⁢              1              ⁢                                                          ⁢              is              ⁢                                                          ⁢              LOW                        ⁢                                                  ⁢                                                  =                          1              ⁢                                                          ⁢              when              ⁢                                                          ⁢              PH              ⁢                                                          ⁢              1              ⁢                                                          ⁢              is              ⁢                                                          ⁢              HIGH                                                          (        4        )            
Since PH1 and PH2 are non-overlapping clocks, equation (4) is re-written as
                                          V            neq                    ⁡                      (            t            )                          =                                            V              n                        ⁡                          (              t              )                                +                                    (                                                R                  i                                ⁢                                  C                  i                                            )                        ⁢                                          ⅆ                                                                                              ⅆ                t                                      ⁢                                          V                n                            ⁡                              (                t                )                                              +                                    U              ⁡                              (                                  PH                  ⁢                                                                          ⁢                  1                                )                                      ×                          (                                                                    V                    n                                    ⁡                                      (                    t                    )                                                  ×                                                      C                    i                                    /                                      C                    1                                                              )                                                          (        5        )            
By analyzing equation (5), it is observed that total equivalent noise has two components:
First component,
                    V        n            ⁡              (        t        )              +                  (                              R            i                    ⁢                      C            i                          )            ⁢                        ⅆ                                                          ⅆ          t                    ⁢                        V          n                ⁡                  (          t          )                      ,is a linear function of Vn(t) and its derivative. Hence, if the noise has any base band component it will remain in ADC baseband and out of band component will remain out of band.)
But the component U(PH1)×(Vn(t)×Ci/Ci) is effectively the convolution of two signals Vn(t) and clock signal PH1 in the frequency domain.
The spectrum of Vn(t) convolves with spectrum of clock PH1 which results in out of band frequencies folding back in the ADC baseband.
If the frequency of clock signal during PH1 is f0 and W is a frequency less than the maximum base band frequency, then any noise present in Vn(t) at frequency f0+W would fold back to the base band frequency W.